Hello,
Naoto Shirogane that rule pertains to guilds, but clubs are similar.
Yes, you can have links to your chat in the club thread, as well as in your sig, but posting it anywhere else is advertising. PMs are permissible as a one-time PM to current club members, or if they ask you for the link (same for VMs on the latter).
It would be best to leave chat stuff in the chatroom. The reasoning being, if there is an argument/problem in the chatroom (or about what happened in the chatroom), it is not Serebii's mods jurisdiction, so you need to handle it outside of Serebii (with the chat site's staff if necessary).